Best of luck with your application, hoping you hear the good news soon!
Firstly, teaching quality. I can only really speak about econ, but I do know friends who have problems with teaching in departments such as International Relations, Statistics, etc. So far, I can confidently that teaching quality, at least for the first 2 years of econ is amazing! I have no problems with the way any of my lecturers deliver the course, I do prefer some over others of course (shoutout Dimitra Petropoulou for EC201(micro) and Ricardo Reis for EC210(macro) haha) but overall lecturers are very experienced, and can explain things very intelligently. This is quite important because simply holding degrees from great universities does not mean you're fit to teaching students. However, for econ, lecturers are intelligent and genuinely passionate about what they do, they're always open to questions and offer a lot of support during their office hours. However, regarding class teachers and research students, you can end up with someone who is quite intelligent of course, but simply cannot deliver the content and manage the class confidently. But then again, classes are simply for going over problem set questions, and we do overcome such a problem by going to another (better) teacher's or even the lecturer's office hour. Like I said in the beginning, I did hear about problems with teaching in other departments, which could possibly explain the low rating in student satisfaction, but I can definitely say there are no fundamental issues within the econ department
In terms of societies, I know people who are active members in about 3 or 4 societies and normal members in another 3 whilst still maintaining excellent grades. By active I mean, holding positions of responsibility and leadership. To be honest, this all depends on how you manage your time and how quickly and efficiently you can do the academic side of university. But, the short answer is, yes its possible to be a part of a sports team and plenty of societies and still have solid grades.
There are many societies that are very active and big on campus (now online) such as the Economics Society, Consultancy society, Business and Investment Group society, Entrepreneurs society, Model United Nations, Debating society and of course you have more fun things such as Drama and Music and Cooking societies. It's really very diverse and they do hold really great events that require a lot of effort and planning, and I almost guarantee that whatever you're interested in, you're bound to find a society with people that have similar interests. Hence, joining societies that you're genuinely interested in is probably your safest and most obvious route to making strong friendships.
Personally, I've never lived in halls. But, yes the four most sociable and fun ones are Bankside, Roseberry, Passfield, and Carr-Saunders. They hold a lot of social events and a hall such as Bankside has a bar, where students can spend a lot of time getting to know each other outside university. My suggestion is, that LSE halls tend to overall be more enjoyable than Uni of London halls and living outside campus, for the obvious reason that these are all people that you potentially have so much in common with, and over time, given that you'll be doing so much together, you're more likely to develop friendships with the people in your halls. But in 2nd and 3rd year, the majority of people do tend to rent flats with their friends which is obviously cheaper and perhaps, nicer, because now you're only sharing a space with people you're comfortable with.
I don't really have much to say on the gender ratio, I don't feel like there's any noticeable difference in the numbers really and so this will not be a problem at all. However, LSE is very known for the large number and portion of international students that study at university. It's very international and while you have people from so so many different parts and countries of the world, you do see a considerable large fraction of Asians (China and Malaysia in particular) among students, this is especially the case in more quantitative courses that require a lot of maths. But not so much in qualitative ones really. This does have its advantages, such as the fact that you'll get to meet people coming from different cultures and backgrounds but this can sometimes not be the greatest thing, because students from certain countries do tend to stick together and don't really interact with a lot of 'outsiders.' But it's worth mentioning, that this isn't that big of an issue because generally, everyone is so keen on making friends from different backgrounds like I said, that you won't really care or bother with those that aren't
I can say that this is something that students definitely do and it does make a difference of course. There are plenty of courses that are easier to study than others and if students feel that their course load is already quite heavy then they do choose something a little light and easy for their outside option. Personally, I have chosen to do a particularly challenging maths course in the first year (MA103 - abstract maths, do check it out!) and because I found it to be quite hard, I chose something lighter in the second year because I felt like my econ courses were going to be really demanding and I couldn't go through that with my outside option again. It doesn't really make a difference except if you're planning to move on to grad studies. You'd be better off choosing modules that can prepare you and are more relevant to your subject of study, instead of just opting for an easy option. But even then, it doesn't really make that big of a difference really.
Finally, I would say no matter how much you prepare and plan for your experience at LSE, you'll never really expect what's coming haha. One thing that I would definitely keep in mind before starting at LSE, is to simply try and do my best in every aspect really but never compare myself to those around me. At school, we're all naturally one of the highest achieving and most hard-working students but at LSE, you're suddenly the average student. There will be times where you struggle while others around you seem to be achieving and doing great things, that maybe be the case, or it's just simply the fact that the grass is greener on the other side. Either way, keep in mind you're smart, hard-working, and intelligent, and don't let the environment trick you into thinking otherwise. Believe in yourself, and do not compare yourself to others